Question

Find the area of the shaded portion of the figure. Each vertex of square ABCD is at the center of a circle. Round your answer to the nearest hundredth.

Answer 1

Given:

Each circle has a diameter of 2 inches each.

The outer square has a side length of 4 inches and the square ABCD has a side length of 2 inches.

To find:

The area of the shaded region.

Solution:

Each circle has a diameter of 2 inches. The square ABCD is at the center of each circle so it has a side length of 1 inch.

To determine the area of the shaded region, we subtract the area of the quarter-circles in the square ABCD from the area of the square ABCD.

The area of a quarter-circle $=\frac{\pi r^{2} }{4} .$

All the quarter-circles have a radius of 1 inch.

The area of 1 quarter-circle $=\frac{\pi r^{2} }{4} = \frac{\pi (1^{2}) }{4} = \frac{3.1415}{4} .$

The area of 4 quarter-circles $=4(\frac{3.1415}{4}) = 3.1415.$

So the area of the quarter-circles in the square ABCD is 3.1514 square inches.

The area of a square $= a^{2} .$

The area of square ABCD $=2^{2} =4.$

The area of the shaded region $=4-3.1415=0.8585.$

The area of the shaded region is 0.8585 square inches.